Learn Russian: Вчера я обошёл эту огромную лужу. - Yesterday I went around this huge puddle.

The prefix об- (around/avoiding) changes to обо- before certain clusters of consonants to make it easier to pronounce. The base verb here is the perfective пойти / идти (to go on foot). Its past tense is irregular: шёл, шла, шло, шли. When you add the prefix обо- to the masculine past tense шёл, you get обошёл.

Why is "this huge puddle" (эту огромную лужу) in the accusative case? Don't we need a preposition for "around"?

When you use a verb with the об-/обо- prefix meaning "to go around" or "to avoid" a physical obstacle, it takes a direct object in the accusative case. The "around" meaning is already built right into the verb's prefix, so you don't need a separate preposition (like вокруг).

Could I use the imperfective verb обходил instead of обошёл here?

Using обходил would change the meaning. Because обошёл is perfective, it means you successfully went around the puddle one specific time and completed the action ("Yesterday I went around it"). If you used обходил, it would imply a repeated habit ("I used to go around it") or focus on the process of doing it.

Why does the adjective огромную end in -ую while the noun лужу ends in just -у?